Article excerpt

There have been a number of efforts to incorporate instructional design principles into mathematics teaching in order to improve student achievement. For instance, an instructional design approach including a mastery learning strategy and incorporating the events of instruction has been used to modify mathematics instruction in Malaysia (Hashim & Tik, 1997). Other research has examined the role of the teacher in mathematics learning and recent findings indicate that students' enjoyment of geometry increased while in an open-ended learning environment where the teacher served as a facilitator (Hannafin, Burruss, & Little, 2001). Instructional design strategies have been employed to develop a video-based series that supports students' ability to solve complex problems and to apply mathematical thinking to other academic subjects (Cognition and Technology Group at Vanderbilt, 1992, 1993). Similarly, instructional software based on models of numerical operations was found to enable students to progress from whole numbers to being able to use fractions (Olive, 2002). Finally, the use of open-ended mathematics problems that included cooperative learning resulted in improved learning and higher confidence levels for at-risk students (Robert, 2002). Consequently, it is apparent that designing effective instructional programs for mathematics is critical for improving student achievement.

Considerable research has examined mathematics classrooms in Japan and several teaching strategies used for mathematics instruction have been identified. For instance, a qualitative analysis of a typical fifth-grade lesson in Japan indicated that an emphasis was placed on the problem-solving processes used by students and errors were discussed in order to examine the breadth of strategies that were used (Sawada, 1999). Further, there was an extensive use of objects that could be manipulated and there was considerable time spent on a single problem in order to provide time for students to examine multiple approaches to problem-solving and to compare multiple solutions (Sawada, 1999). Similarly, seventh-grade students in Japan who were able to exhibit divergent thinking strategies showed a greater number of solutions to open-ended mathematics problems (Imai, 2000). Other studies of mathematics classrooms in Japan have found that extensive time was spent on verbal explanations of specific problems and their solutions (Becker, Silver, Kantowski, Travers, & Wilson, 1990) and that students in Japan spent considerably more time on homework activities than did students in the United States (Stigler, Lee, Lucker, & Stevenson, 1982). In addition, Japanese teachers more often directed their time toward discussing questions with the entire class rather than with individual students (Stigler, Lee, & Stevenson, 1987). Results from the TIMSS Videotape Classroom Study indicated that students in eighth-grade mathematics classrooms in Japan also spent considerably more time on inventing and developing multiple problem-solving solutions to geometry problems than did students in the United States (Stigler, Gallimore, & Hiebert, 2000). Other research suggests that students in Japan tended to provide solutions that used higher levels of mathematical sophistication (Silver, Leung, & Cai, 1995). Finally, cross-cultural comparisons from the initial TIMSS assessment revealed that eighth-grade mathematics instruction in Japan covered fewer topics, but in greater detail, than was the case for United States mathematics classrooms (National Research Council, 1999).

The importance of homework activities for student achievement has been extensively discussed. Research has indicated that, in general, students who spend more time on homework tend to show higher levels of academic achievement (Cooper & Valentine, 2001). For instance, results from the Third International Mathematics Study (TIMSS) indicated that adolescent students in Japan who showed higher mathematics achievement test scores reported that their teachers more frequently gave them homework (House, 2001). …

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